This thesis investigates weak lensing (WL) of galaxies and the cosmic microwave back- ground (CMB) in the nonlinear regime. WL describes the effect of bending of background light rays by foreground matter (“lens”). It is sensitive to the large scale structure of the universe, and hence is a promising method to answer some unsolved fundamental questions in physics, such as the nature of dark energy and the total mass of neutrinos. WL datasets of unprecedented precision will come on-line in the early 2020s. This presents an exciting yet challenging task for the WL community: how do we extract the maximum amount of information from lensing observables, while minimizing the impact of systematics? This work attempts to answer this question by studying non-Gaussian statistics. Traditionally, WL data are analyzed using second-order statistics, which capture all the cosmological information if the density field is Gaussian. However, the small-scale density fluctuations are strongly non-Gaussian and can be highly sensitive to cosmology.

Thus we need higher order (non-Gaussian) statistics to utilize these features in the nonlinear regime. In this thesis, we study the constraining power on cosmology and relevant systematics of non-Gaussian statistics, with a focus on convergence peaks. We present the first cosmological constraints using peak counts of the CFHTLenS survey. We also quantify the impact of magnification and size bias, one type of lensing systematics, on the lensing power spectrum and peaks. Finally, going beyond galaxy lensing, we cross-correlate Planck CMB lensing maps with CFHTLenS galaxy lensing maps, to investigate various WL systematics.